Nilpotent group

 ( Nilpotent Groups ) 

In mathematics, specifically group theory, a nilpotent group G is a group that has an upper central series that terminates with G. Equivalently, it has a central series of finite length or its lower central series terminates with {1}.

Intuitively, a nilpotent group is a group that is "almost Abelian group". This idea is motivated by the fact that nilpotent groups are Solvable group, and for finite group nilpotent groups, two elements having relatively prime orders must commute. It is also true that finite nilpotent groups are supersolvable. The concept is credited to work in the 1930s by Russian mathematician Sergei Chernikov.

Nilpotent groups arise in Galois theory, as well as in the classification of groups. They also appear prominently in the classification of .

Analogous terms are used for (using the Lie bracket) including nilpotent, lower central series, and upper central series.

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Definition The definition uses the idea of a central series for a group. The following are equivalent definitions for a nilpotent group :

For a nilpotent group, the smallest such that has a central series of length is called the nilpotency class of ; and is said to be nilpotent of class . (By definition, the length is if there are $n\; +\; 1$ different subgroups in the series, including the trivial subgroup and the whole group.)

Equivalently, the nilpotency class of equals the length of the lower central series or upper central series. If a group has nilpotency class at most , then it is sometimes called a nil- group.

It follows immediately from any of the above forms of the definition of nilpotency, that the trivial group is the unique group of nilpotency class , and groups of nilpotency class  are exactly the non-trivial abelian groups.

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Examples

• As noted above, every abelian group is nilpotent.
• For a small non-abelian example, consider the quaternion group Q₈, which is a smallest non-abelian p-group. It has center {1, −1} of order 2, and its upper central series is {1}, {1, −1}, Q₈; so it is nilpotent of class 2.
• The direct product of two nilpotent groups is nilpotent.
• All finite p-group are in fact nilpotent (proof). For n > 1, the maximal nilpotency class of a group of order p ⁿ is n - 1 (for example, a group of order p ² is abelian). The 2-groups of maximal class are the generalised , the , and the semidihedral groups.
• Furthermore, every finite nilpotent group is the direct product of p-groups.
• The multiplicative group of upper unitriangular n × n matrices over any field F is a nilpotent group of nilpotency class n − 1. In particular, taking n = 3 yields the Heisenberg group H, an example of a non-abelian infinite nilpotent group. It has nilpotency class 2 with central series 1, Z( H), H.
• The multiplicative group of Borel subgroup n × n matrices over a field F is not in general nilpotent, but is solvable group.
• Any nonabelian group G such that G/ Z( G) is abelian has nilpotency class 2, with central series {1}, Z( G), G.

The k for which any group of order k is nilpotent have been characterized .

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Explanation of term Nilpotent groups are called so because the "adjoint action" of any element is nilpotent, meaning that for a nilpotent group $G$ of nilpotence degree $n$ and an element $g$, the function $\backslash operatorname\{ad\}\_g\; \backslash colon\; G\; \backslash to\; G$ defined by $\backslash operatorname\{ad\}\_g(x)\; :=\; g,x$ (where $g,x=g^\{-1\}\; x^\{-1\}\; g\; x$ is the commutator of $g$ and $x$) is nilpotent in the sense that the $n$th iteration of the function is trivial: $\backslash left(\backslash operatorname\{ad\}\_g\backslash right)^n(x)=e$ for all $x$ in $G$.

This is not a defining characteristic of nilpotent groups: groups for which $\backslash operatorname\{ad\}\_g$ is nilpotent of degree $n$ (in the sense above) are called $n$-,For the term, compare Engel's theorem, also on nilpotency. and need not be nilpotent in general. They are proven to be nilpotent if they have finite order, and are to be nilpotent as long as they are finitely generated.

An abelian group is precisely one for which the adjoint action is not just nilpotent but trivial (a 1-Engel group).

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Properties Since each successive factor group in the central series is abelian, and the series is finite, every nilpotent group is a solvable group with a relatively simple structure.

Every subgroup of a nilpotent group of class is nilpotent of class at most ;Bechtell (1971), p. 51, Theorem 5.1.3 in addition, if is a homomorphism of a nilpotent group of class , then the image of is nilpotent of class at most .

Statement (d) can be extended to infinite groups: if is a nilpotent group, then every Sylow subgroup of is normal, and the direct product of these Sylow subgroups is the subgroup of all elements of finite order in (see torsion subgroup).

Many properties of nilpotent groups are shared by hypercentral groups.

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Notes

• (2026). 9783110156294, Walter de Gruyter. ISBN 9783110156294
• (1974). 9780387905181, Springer-Verlag. ISBN 9780387905181
• I. Martin Isaacs (2026). 9780821843444, American Mathematical Society. ISBN 9780821843444
• Palmer, Theodore W. (1994). 9780521366380, Cambridge University Press. ISBN 9780521366380
• review
• Suprunenko, D. A. (1976). 9780821813416, American Mathematical Society. ISBN 9780821813416
• (2026). 9781852332358, Springer. ISBN 9781852332358
• (1999). 9780486409221, Dover Publications. ISBN 9780486409221

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